Extending homeomorphisms in compactification of Fréchet spaces

In the study of extending homeomorphisms in the compactification of Fréchet spaces into the Hilbert cube $Q$, it is shown that for a given homeomorphism $f$ of $s$ onto itself and a closed subset $K$ of $s$ satisfying property $Z$ in $s$, there is a homeomorphism $g$ of $s$ onto itself such that $gf{g^{ - 1}}{|_{g(K)}}$ extends to a homeomorphism of $Q$ onto $Q$. The proof is by employing the rather useful shifting homeomorphism on $Q$.